On the divisibility of matrices associated with multiplicative functions

Suppose that $n$ and $k$ are positive integers. Let $S=\{x_1,\dots,x_n\}$ be a sequence of $n$ distinct positive integers, and let $f$ be an integer-valued multiplicative function. The sequence $S$ is called a divisor chain if $x_{\sigma(1)}|\dots|x_{\sigma(n)}$ for some permutation $\sigma$ on $\{1,\dots, n\}$. We say that the sequence $S$ consists of $k$ coprime divisor chains if $S$ can be partitioned as the union of $k$ divisor chains $S_1,\dots, S_k$ such that each element of $S_i$ is coprime to each element of $S_j$ for all integers $i$ and $j$ with $1\le i\ne j\le k$. In this paper, we show that for any divisor chain $S$, the matrix $(f(S))$ with entries $f(\gcd(x_i, x_j))$ divides the matrix $(f[S])$ with entries $f({\rm lcm}(x_i, x_j))$ in the ring $M_n({\bf Z})$ of $n\times n$ matrices over the integers if and only if $f(\min(S))|f(x_i)$ for all integers $i\in\{1,\dots, n\}$. This strengthens a result of Hong [14]. For any positive integer $a$ and any sequence $S$ consisting of two coprime divisor chains with $1\not\in S$, we show that the matrix $(f(S^a))$ divides the matrix $(f[S^a])$ in $M_n({\bf Z})$, where $S^a:=\{x_1^a,\dots,x_n^a\}$. This confirms a conjecture of Chen and Hong. We show also that such factorization is no longer true in general if $S$ consists of at least three coprime divisor chains with $1\not\in S$. We conjecture that if $k\ge 3$, then the GCD matrix $(S)$ does not divide the LCM matrix $[S]$ in the ring $M_{k}({\bf Z})$ if $S$ consists of the first $k$ odd prime numbers.